Proof of Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function

Conventions. From the setting we use the real numbers and their order, Euclidean space with its difference of points, norm and distance $d_{E}$, the set $\mathcal{S}(n)$ with its norm and distance $d_{\mathcal{S}(n)}$, and the notions of class $C^{2}$, gradient and Hessian; no matrix ordering is needed. Recall from clause Second-Order Equations on Euclidean Open Sets (iii) that $\mathcal{S}(n)$ is closed under differences and that $d_{\mathcal{S}(n)}(P,Q)=\lVert P-Q\rVert$, and from clause Second-Order Equations on Euclidean Open Sets (ii) that the $j$th coordinate of a difference $z-w$ of points of $\mathbb{R}^{n}$ is $z_{j}-w_{j}$. Elementary order facts are those of Elementary Order Arithmetic in an Ordered Field. We write $\iota$ for the canonical map of $\mathbb{N}$ into $\mathbb{R}$ of Properties of the Canonical Map from the Natural Numbers to an Ordered Field and $\sigma_{n}$ for the sum of $n$ ones, as in One Modulus and a Sum Bound for a Finite Family (ii).

Proof of claim 1. Regard $\psi$ as a map into $\mathbb{R}^{1}$ with single coordinate function $\psi$, as clause 3 of Functions of Class Cᵏ on an Open Subset of ℝⁿ permits. Claim 3 of Euclidean Space Is Open in Itself then states precisely that $\psi$ is continuous relative to $V$ at every point of $V$, as a map from $V$ into $(\mathbb{R},d_{\mathbb{R}})$.

Proof of claim 2. Since $\psi$ is of class $C^{1}$ on $V$, clause 1 of Functions of Class Cᵏ on an Open Subset of ℝⁿ shows that for every $j$ with $1\le j\le n$ the partial derivative of $\psi$ with respect to the $j$th variable exists at every point of $V$ and that $\partial_{j}\psi:V\to\mathbb{R}$ is continuous in the Euclidean sense at every point of $V$; by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions the function $\partial_{j}\psi$ is therefore continuous at every point of $V$ relative to $V$, as a map into $(\mathbb{R},d_{\mathbb{R}})$. In particular $D\psi(y)$ is defined for every $y\in V$.

Let $a\in V$ and let $\varepsilon\in\mathbb{R}$ be positive. The real number $\iota(n)$ is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so $\iota(n)^{-1}$ is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field and $t=\tfrac{\varepsilon}{2}\,\iota(n)^{-1}$ is positive, $\tfrac{\varepsilon}{2}$ being positive by claim 8 of Elementary Order Arithmetic in an Ordered Field. Applying One Modulus and a Sum Bound for a Finite Family (i) to the family of the $n$ functions $\partial_{j}\psi:V\to\mathbb{R}$, each continuous at $a$ relative to $V$ as a map into the metric space $(\mathbb{R},d_{\mathbb{R}})$, we obtain a positive $\delta\in\mathbb{R}$ such that every $y\in V$ with $d_{E}(a,y)<\delta$ satisfies

$$\bigl|\partial_{j}\psi(y)-\partial_{j}\psi(a)\bigr|<t\qquad\text{for every }j\text{ with }1\le j\le n .$$

For such a $y$, the $j$th coordinate of $D\psi(y)-D\psi(a)$ is $\partial_{j}\psi(y)-\partial_{j}\psi(a)$, whose absolute value is at most $t$; hence $\lVert D\psi(y)-D\psi(a)\rVert\le\iota(n)t$ by Coordinate Bounds Control the Euclidean Norm; here $\iota(n)t=\iota(n)\bigl(\tfrac{\varepsilon}{2}\iota(n)^{-1}\bigr)=\tfrac{\varepsilon}{2}$, by the commutativity and associativity of multiplication and the identity $\iota(n)\iota(n)^{-1}=1$ of the ordered field $\mathbb{R}$, and $\tfrac{\varepsilon}{2}<\varepsilon$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, so

$$d_{E}\bigl(D\psi(y),D\psi(a)\bigr)=\lVert D\psi(y)-D\psi(a)\rVert<\varepsilon$$

by the mixed transitivity of claim 2 of Elementary Order Arithmetic in an Ordered Field. As $\varepsilon$ was an arbitrary positive real, the gradient map is continuous at $a$ relative to $V$, and $a\in V$ was arbitrary.

Proof of claim 3. By claim 2 of Basic Properties of Cᵏ Functions the function $\varphi$ is of class $C^{1}$ on $V$, so the assertion about the gradient map is claim 2 applied to $\varphi$.

For the Hessian map, note first that for every $j$ with $1\le j\le n$ the function $\partial_{j}\varphi:V\to\mathbb{R}$ is of class $C^{1}$ on $V$, by clause 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ applied with $k=1$. Let $w_{j}:V\to\mathbb{R}^{n}$ be the map whose value at $y$ is $D(\partial_{j}\varphi)(y)$; by claim 2 it is continuous at every point of $V$ relative to $V$. By the definition of the Hessian matrix and clause 4 of Functions of Class Cᵏ on an Open Subset of ℝⁿ, the entry of $D^{2}\varphi(y)$ in row $i$ and column $j$ is $\partial_{i}\partial_{j}\varphi(y)$, the partial derivative with respect to the $i$th variable of $\partial_{j}\varphi$, which is the $i$th coordinate of $w_{j}(y)$.

Let $a\in V$ and let $\varepsilon\in\mathbb{R}$ be positive. By One Modulus and a Sum Bound for a Finite Family (ii) we have $1\le\sigma_{n}$, so $\sigma_{n}$ is positive, as is $\sigma_{n}^{-1}$ by claim 7 of Elementary Order Arithmetic in an Ordered Field, and

$$t=\tfrac{\varepsilon}{2}\,\sigma_{n}^{-1}\sigma_{n}^{-1}$$

is positive. Applying One Modulus and a Sum Bound for a Finite Family (i) to the family of the $n$ maps $w_{j}:V\to\mathbb{R}^{n}$, each continuous at $a$ relative to $V$ as a map into $(\mathbb{R}^{n},d_{E})$, we obtain a positive $\delta\in\mathbb{R}$ such that every $y\in V$ with $d_{E}(a,y)<\delta$ satisfies $\lVert w_{j}(y)-w_{j}(a)\rVert<t$ for every $j$ with $1\le j\le n$.

Fix such a $y$ and put $X=D^{2}\varphi(y)-D^{2}\varphi(a)$, an element of $\mathcal{S}(n)$. For all $i,j$ with $1\le i\le n$ and $1\le j\le n$, the entry $X_{ij}$ is the difference of the corresponding entries, that is the $i$th coordinate of $w_{j}(y)-w_{j}(a)$, so claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square gives

$$|X_{ij}|\le\lVert w_{j}(y)-w_{j}(a)\rVert<t ,$$

hence $|X_{ij}|\le t$. By One Modulus and a Sum Bound for a Finite Family (iii), applied for each fixed $i$ to the $n$-tuple whose $j$th component is $|X_{ij}|$,

$$\sum_{j=1}^{n}|X_{ij}|\le\sigma_{n}t ,$$

and applying the same claim to the $n$-tuple whose $i$th component is $\sum_{j=1}^{n}|X_{ij}|$, together with claim 5 of Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix, gives

$$\lVert X\rVert\le\sum_{i=1}^{n}\sum_{j=1}^{n}|X_{ij}|\le\sigma_{n}\bigl(\sigma_{n}t\bigr)=\tfrac{\varepsilon}{2}<\varepsilon .$$

Here $\sigma_{n}(\sigma_{n}t)=\tfrac{\varepsilon}{2}$ by the commutativity and associativity of multiplication and the identity $\sigma_{n}\sigma_{n}^{-1}=1$ of the ordered field $\mathbb{R}$, and $\tfrac{\varepsilon}{2}<\varepsilon$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, so $\lVert X\rVert<\varepsilon$ by the mixed transitivity of claim 2 of that lemma. Since $d_{\mathcal{S}(n)}\bigl(D^{2}\varphi(y),D^{2}\varphi(a)\bigr)=\lVert X\rVert$, and $\varepsilon$ was an arbitrary positive real, the Hessian map is continuous at $a$ relative to $V$; and $a\in V$ was arbitrary. $\blacksquare$